Mathematical Physics I

Question 1

n-th order ODE

Answer 1

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Question 2

solution of nth order ODE

Answer 2

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Question 3

autonomous

Answer 3

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Question 4

initial value problem

Answer 4

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Question 5

solution to IVP

Answer 5

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Question 6

matrix exponential

Answer 6

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Question 7

fixed point

Answer 7

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Question 8

Picard-Lindeloef Theorem (versions)

Answer 8

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Question 9

prolongation of solutions

Answer 9

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Question 10

maximal solution

Answer 10

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Question 11

phase space

Answer 11

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Question 12

dynamical systems

• invertible (G is what?)
• continuous
• discrete

Answer 12

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Question 13

flow of IVP

Answer 13

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Question 14

lie group

Answer 14

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Question 15

orbit

Answer 15

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Question 16

cycle

Answer 16

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Question 17

equilibrium point

Answer 17

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Question 18

period

Answer 18

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Question 19

phase portrait

Answer 19

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Question 20

invariant set/function

Answer 20

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Question 21

Lyapunov stable

Answer 21

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Question 22

attracting

Answer 22

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Question 23

asymptotically stable

Answer 23

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Question 24

Lie theorem

Answer 24

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Question 25

infinitesimal transformation

Answer 25

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Question 26

Lie derivative of F along v

Answer 26

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Question 27

Theorem 2.3

Answer 27

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Question 28

Theorem 2.4

Answer 28

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Question 29

phase volume

Answer 29

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Question 30

Liouville Theorem

Answer 30

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Question 31

Poincare Theorem

Answer 31

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Question 32

Lyapunov function (strict)

Answer 32

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Question 33

Lyapunov Theorem

Answer 33

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Question 34

linear autonomous and homogeneous IVP

Answer 34

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Question 35

exponential matrix

Answer 35

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Question 36

general linear Lie algebra

Answer 36

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Question 37

Jordan normal form

Answer 37

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Question 38

generalized eigenspace

Answer 38

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Question 39

Jordanization of A

Answer 39

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Question 40

real Jordan blocks

Answer 40

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Question 41

Theorem 2.10

Answer 41

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Question 42

stable, unstable, center manifold

Answer 42

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Question 43

hyperbolic fixed point

Answer 43

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Question 44

Theorem 2.13

Answer 44

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Question 45

linearization of IVP

Answer 45

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Question 46

topologically equivalent

Answer 46

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Question 47

topologically conjugate

Answer 47

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Question 48

orbitally equivalent

Answer 48

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Question 49

Hartmann-Gromwell Theorem

Answer 49

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Question 50

standard saddle system

Answer 50

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Question 51

fundamental solution

Answer 51

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Question 52

Theorem (Abel-Liouville)

Answer 52

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Question 53

non-antonomous, non homogeneous linear IVP

Answer 53

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Question 54

periodic homogeneous IVP

Answer 54

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Question 55

monodromy matrix

Answer 55

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Question 56

Floquet multipliers

Answer 56

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Question 57

Floquet theorem

Answer 57

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Question 58

bifurcations

Answer 58

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Question 59

topological normal form

Answer 59

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Question 60

genericity conditions

Answer 60

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Question 61

topological equivalence

Answer 61

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Question 62

bifurcation point

Answer 62

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Question 63

bifurcation diagram

Answer 63

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Question 64

saddle node bifurcations

Answer 64

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Question 65

Hopf bifurcations

Answer 65

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Question 66

configuration space

Answer 66

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Question 67

mechanical system

Answer 67

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Question 68

Newton equations

Answer 68

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Question 69

potential energy

Answer 69

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Question 70

conservative system

Answer 70

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Question 71

kinetic energy

Answer 71

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Question 72

Lagrangian

Answer 72

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Question 73

differentiable functional

Answer 73

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Question 74

extremal

Answer 74

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Question 75

action functional

Answer 75

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Question 76

Lemam 3.1

Answer 76

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Question 77

Euler-Lagrange equations

Answer 77

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Question 78

Legendre condition

Answer 78

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Question 79

symmetry Lie group

Answer 79

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Question 80

momentum conjugated to p

Answer 80

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Question 81

Legendre transformation

Answer 81

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Question 82

canonical hamiltonial

Answer 82

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Question 83

divergence free

Answer 83

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Question 84

symplectic matrix

Answer 84

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Question 85

canonical symplectic vector space

Answer 85

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Question 86

symplectic Lie algebra

Answer 86

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Question 87

canonical Poisson bracket

Answer 87

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Question 88

poisson involution

Answer 88

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Question 89

canonical H. equations

Answer 89

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Question 90

canonical H. flow.

Answer 90

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Question 91

canonical Poisson structure

Answer 91

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Question 92

Poisson Theorem

Answer 92

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Question 93

canonical transformation

Answer 93

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Question 94

symplectic transformation

Answer 94

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Question 95

wedge product

Answer 95

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Question 96

differential n-form

Answer 96

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Question 97

volume form

Answer 97

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Question 98

exterior derivative

Answer 98

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Question 99

exactness property

Answer 99

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Question 100

canonical symplectic 2-form

Answer 100

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Question 101

generating functions

Answer 101

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Question 102

smooth distribution

Answer 102

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Question 103

integral manifold

Answer 103

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Question 104

integrable distribution

Answer 104

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Question 105

Froebenius Theorem

Answer 105

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Question 106

k-vector fields

Answer 106

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Question 107

Lie derivative of a 2-vector field

Answer 107

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Question 108

Cartan formula

Answer 108

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Question 109

Lie algebra

Answer 109

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Question 110

matrix Lie group

Answer 110

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Question 111

Poisson bracket

Answer 111

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Question 112

Poisson manifold

Answer 112

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Question 113

Hamiltonian VF

Answer 113

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Question 114

Casimir function

Answer 114

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Question 115

Poisson involution

Answer 115

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Question 116

rank of P. manifold

Answer 116

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Question 117

regular P. manifold

Answer 117

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Question 118

symplectic manifold

Answer 118

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Question 119

symplectic morphism

Answer 119

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Question 120

symplectic foliation dim m

Answer 120

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Question 121

Euler top

Answer 121

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